NP-completeness of the ${\mathrm{\ell }}_1$-embedding problem for simple graphs as sphere-of-influence graphs

In graph theory an interesting question is whether for a fixed choice of $p\in [0,\mathrm{\infty }]$, all simple graphs appear as sphere-of-influence graphs in some Euclidean space with respect to the ${\mathrm{\ell }}_p$ metric. The answer is affirmative for $p=\mathrm{\infty }$, negative for any $p\in (0,\mathrm{\infty })$, and unknown for $p=1$. The result of this work shows that for the case of $p=1$, this embeddability question is a (Promise) NP-Complete problem.